We refer the interested readers to more complete presentations for the case
where the energy is derived from an energy functional [1]}. When
the effective Hamiltonian reads \(\hat{H} = \hat{T} + \hat{V}\)  with \(\hat{V}\) 
containing only two-body effective forces, the total energy becomes
\(E[\rho ,\kappa ,\kappa ^{*}] =\sum _{ij} t_{ij} \rho _{ji}+\frac{1}{2} \sum _{ijkl} \bar{v}_{ijkl}\rho _{lj}\rho _{ki}+\frac{1}{4}\sum _{ijkl} \bar{v}_{ijkl} \kappa ^{*}_{ij}\kappa _{kl}.\) 
